FRM Exam Part I · Measures of Financial Risk

Standard Deviation and Other Risk Measures Compared

Updated 11 October 2026 · Fact-checked

Standard deviation measures the dispersion of returns around the mean, up and down alike. Downside measures look only at losses, scenario measures test specific events, and quantile measures such as VaR give a loss threshold at a confidence level. To solve questions, match the measure to what it captures and what it ignores.

Understand Standard Deviation and Other Risk Measures

Every risk measure compresses a whole distribution of outcomes into one number. Each one keeps some information and throws the rest away. The exam tests what each measure keeps and what it ignores.

Standard deviation (volatility) is the square root of the average squared deviation from the mean. It is simple, easy to scale across time and easy to combine across assets using correlations. But it treats gains and losses the same. It is also a complete description of risk only when returns are roughly normal or elliptical. With skewed or fat-tailed returns, two portfolios can share the same standard deviation and have very different tail losses.

Downside measures fix the symmetry problem. Semi-deviation uses only deviations below the mean (or below a target). Downside deviation uses deviations below a minimum acceptable return. Shortfall probability is the chance of falling below a target. These suit investors who care about losses, not upside surprises.

Scenario-based measures ask what happens in a specified event, such as a 200 basis point rate rise or an equity fall of 30%. Stress tests and scenario analysis need no distribution assumption and can capture events absent from history. Their weakness is subjectivity: the result depends on which scenarios you pick, and they usually carry no probability.

Quantile-based measures read a point in the loss distribution. VaR is the loss that will not be exceeded with probability equal to the confidence level over a horizon. It says nothing about how bad losses are beyond that point, and it is not always subadditive. Expected shortfall (ES) is the average loss in the tail beyond VaR. It is subadditive and so coherent, but it is harder to estimate and backtest.

Key formulas to remember

Standard deviation: σ = √[ Σ(Rᵢ − R̄)² ÷ (n − 1) ]; Sample version uses n − 1. Symmetric: gains and losses count equally.
Downside deviation: DD = √[ Σ min(Rᵢ − MAR, 0)² ÷ n ]; Only returns below the minimum acceptable return (MAR) contribute. Divide by the total number of observations, not just the bad ones, unless the question says otherwise.
Normal VaR: VaR = −μ + z × σ (as a loss); z = 1.645 at 95% and 2.326 at 99% (one-tailed). Valid only under normality.
Horizon scaling: σ(T days) = σ(1 day) × √T; Assumes independent returns with constant volatility.
Expected shortfall: ES = E[ Loss | Loss ≥ VaR ]; Always at least VaR at the same confidence level.
Coherence: Monotonicity, translation invariance, positive homogeneity, subadditivity; ES is coherent. VaR fails subadditivity in general. Standard deviation fails monotonicity.

How to solve Standard Deviation and Other Risk Measures questions

Use this approach for any question that compares risk measures or asks you to compute one.

1Identify which measures the question names and what it asks: compute, compare or criticise.
2Recall what each measure captures: dispersion, losses only, a specific event, or a tail point.
3Check the distribution: is it normal, skewed, or fat-tailed? That decides whether standard deviation or VaR is reliable.
4If computing, write the formula, plug in the numbers, and keep units (percent or currency) consistent.
5Scale for horizon or confidence only if the question's assumptions allow it.
6State the limitation that matters here: symmetry, no tail information, subjectivity, or non-subadditivity.
7Eliminate options that overstate a rule, such as saying VaR is always coherent.

Quickest way: Match measure to its blind spot

When to use it: For conceptual comparison questions with four options.

Standard deviation: blind to direction and tail shape.
Semi-deviation: blind to upside and extreme tails.
Scenarios: blind to probability and unchosen events.
VaR: blind to loss size beyond the threshold, and can fail subadditivity.
ES: sees the tail, but needs more data and is harder to backtest.
Pick the option that names the correct blind spot for the measure in the question.

Common mistakes in Standard Deviation and Other Risk Measures

Saying standard deviation measures only downside risk.
Candidates equate volatility with losses in everyday language.
Fix: Remember it counts upside deviations too. Use semi-deviation for downside only.
Treating VaR as the maximum possible loss.
The word 'worst' appears in many VaR definitions.
Fix: VaR is a threshold at a confidence level. Losses beyond it can be far larger.
Claiming VaR is coherent.
It is widely used, so people assume it is well-behaved.
Fix: VaR can violate subadditivity. ES is the coherent alternative.
Assuming scenario analysis gives a probability.
It is mixed up with quantile measures.
Fix: Scenarios give the loss in a defined event. Probability is usually not attached.
Using the wrong z-value or the two-tailed value.
Tables show both one- and two-tailed values.
Fix: VaR is one-tailed: 1.645 for 95% and 2.326 for 99%.
Applying √T scaling when returns are autocorrelated or volatility changes.
The rule is taught as a general formula.
Fix: State the assumptions: independent returns and constant volatility.

Worked examples

Example 1

A portfolio has a daily return standard deviation of 1.2% and a mean of zero. Using the normal distribution, what is the 10-day 99% VaR as a percentage of portfolio value? (z = 2.326)

Show the solution

Scale volatility: 1.2% × √10 = 1.2% × 3.1623 = 3.7947%.
Apply VaR = z × σ − μ with μ = 0: 2.326 × 3.7947% = 8.826%.
Round to two decimals.

Answer: About 8.83% of portfolio value

Example 2

Portfolio A and Portfolio B both have annual standard deviation of 10%. B has a small chance of a very large loss, while A's returns are normal. Which statement is most accurate? (a) Both have the same risk because the standard deviations are equal. (b) B has greater tail risk than standard deviation shows. (c) A has greater tail risk because it is normal. (d) Standard deviation is always larger for fat-tailed returns.

Show the solution

Standard deviation summarises dispersion only, not the shape of the tails.
B's rare large loss means a fatter left tail than a normal distribution with the same σ.
So tail measures such as VaR at a high confidence level or ES would show B as riskier.
Option (a) ignores the tail. Option (c) is reversed. Option (d) overstates: there is no such rule.

Answer: (b) B has greater tail risk than standard deviation shows

Exam tips

Expect comparison questions: pick the option that names the correct limitation, and watch for words like 'always' and 'only'.
Memorise z-values 1.645 and 2.326 and the √T rule, and write the formula before calculating.
Know that ES is coherent and VaR is not in general, and why this matters for aggregation.
For scenario and stress-test questions, remember the strengths: no distribution needed and can capture unseen events. The weakness is subjectivity and no probabilities.
On a financial calculator, use the √ key for horizon scaling, and keep four decimals until the final answer.

Practice questions from Measures of Financial Risk

Standard Deviation and Other Risk Measures in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Standard Deviation and Other Risk Measures: frequently asked questions

What are the main limitations of standard deviation as a risk measure?

It treats gains and losses equally and says nothing about skewness or tail shape. It fully describes risk only for roughly normal returns. It also depends on the sample period and on the estimation method.

What is the difference between standard deviation and VaR?

Standard deviation measures overall dispersion around the mean. VaR is a loss threshold at a chosen confidence level and horizon. VaR focuses on the tail, while standard deviation looks at the whole distribution.

Is stress testing a risk measure for FRM Part I?

Yes, as a scenario-based measure. It estimates losses under specified adverse events and needs no distribution assumption. Its results depend on how scenarios are chosen and usually carry no probability.

Why is expected shortfall preferred over VaR in some frameworks?

ES averages losses beyond VaR, so it reflects tail severity. It is subadditive, which makes it coherent. VaR ignores losses beyond its threshold and can fail subadditivity.